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Require Import
        HoTT.Basics
        HoTT.DProp
        HoTT.BoundedSearch
        HoTT.Types.Universe
        HoTT.Types.Sum
        HoTT.Types.Arrow
        HoTT.Spaces.Finite
        HoTT.ExcludedMiddle.

Require Import
        HoTT.Classes.interfaces.abstract_algebra
        HoTT.Classes.interfaces.orders
        HoTT.Classes.interfaces.integers
        HoTT.Classes.interfaces.rationals
        HoTT.Classes.interfaces.naturals
        HoTT.Classes.interfaces.cauchy
        HoTT.Classes.interfaces.archimedean
        HoTT.Classes.interfaces.round
        HoTT.Classes.interfaces.naturals
        HoTT.Classes.implementations.peano_naturals
        HoTT.Classes.orders.orders
        HoTT.Classes.orders.archimedean
        HoTT.Classes.orders.rings
        HoTT.Classes.orders.dec_fields
        HoTT.Classes.orders.semirings
        HoTT.Classes.orders.lattices
        HoTT.Classes.theory.rings
        HoTT.Classes.theory.apartness
        HoTT.Classes.theory.rationals.

(* Strangely, it seems that combining the next import with the above list breaks some instance search? *)
Require Import
        HoTT.Classes.orders.fields
        HoTT.Classes.theory.fields
        HoTT.Classes.theory.dec_fields.

Local Open Scope type_scope.

Section locator.
  Context (Q : Type).
  Context `{Qrats : Rationals Q}.
  Context {Qdec_paths : DecidablePaths Q}.
  Context {Qtriv : TrivialApart Q}.
  Context `{!Trichotomy (<)}.
  Context (F : Type).
  Context `{Forderedfield : OrderedField F}.
  Context {Fabs : Abs F}.
  Context {Farchimedean : ArchimedeanProperty Q F}.
  Context {Fcomplete : IsComplete Q F}.
  Context {Qroundup : RoundUpStrict Q}.

  Context `{Funext} `{Univalence}.

  (* Assume we have enumerations of the rationals, and of pairs of ordered rationals. *)
  Context (Q_eq : nat <~> Q).
  Context (QQpos_eq : nat <~> Q * Qpos Q).

  Instance qinc : Cast Q F := rationals_to_field Q F.
  (* TODO The following two instances should probably come from the `Rationals` instance. *)
  Context (cast_pres_ordering : StrictlyOrderPreserving qinc)
          (qinc_strong_presving : IsSemiRingStrongPreserving qinc).
  Existing Instance cast_pres_ordering.
  Existing Instance qinc_strong_presving.

  (* Definition of a locator for a fixed real number. *)
  Definition locator (x : F) := forall q r : Q, q < r -> (' q < x) + (x < ' r).

  (* Alternative definition; see equivalence below *)
  Record locator' (x : F) :=
    { locates_right : forall q r : Q, q < r -> DHProp
    ; locates_right_true : forall q r : Q, forall nu : q < r, locates_right q r nu -> ' q < x
    ; locates_right_false : forall q r : Q, forall nu : q < r, ~ locates_right q r nu -> x < ' r
    }.

  Arguments locates_right       [x] _ [q] [r] _.
  Arguments locates_right_true  [x] _ [q] [r] _.
  Arguments locates_right_false [x] _ [q] [r] _.

  Definition locates_left {x : F} (l : locator' x) {q r : Q} : q < r -> DHProp :=
    fun nu => Build_DHProp (Build_HProp (~ (locates_right l nu))) _.

  Section classical.
    Context `{ExcludedMiddle}.

    
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F

locator x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F

locator x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r

(' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r

IsHProp (' q < x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
' q < x -> (' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
~ (' q < x) -> (' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r

IsHProp (' q < x)
apply _.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r

' q < x -> (' q < x) + (x < ' r)
exact inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r

~ (' q < x) -> (' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
notlt:~ (' q < x)

(' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
notlt:~ (' q < x)

x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
H1:ExcludedMiddle
x:F
q, r:Q
ltqr:q < r
notlt:~ (' q < x)
ltqr':' q < ' r

x < ' r
exact (nlt_lt_trans notlt ltqr'). Qed. End classical. Section rational. Context (s : Q).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s:Q

locator (' s)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s:Q

locator (' s)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r

(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltqs:q < s

(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
eqqs:q = s
(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltsq:s < q
(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltqs:q < s

(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltqs:q < s

' q < ' s
apply (strictly_order_preserving _); assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
eqqs:q = s

(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:s < r
eqqs:q = s

(' q < ' s) + (' s < ' r)
apply inr, (strictly_order_preserving _); assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltsq:s < q

(' q < ' s) + (' s < ' r)
apply inr, (strictly_order_preserving _), (transitivity ltsq ltqr); assumption. Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s:Q

locator (' s)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s:Q

locator (' s)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r

(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltsr:s < r

(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
eqsr:s = r
(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltrs:r < s
(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltsr:s < r

(' q < ' s) + (' s < ' r)
apply inr, (strictly_order_preserving _); assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
eqsr:s = r

(' q < ' s) + (' s < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < s
eqsr:s = r

(' q < ' s) + (' s < ' r)
apply inl, (strictly_order_preserving _); assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
s, q, r:Q
ltqr:q < r
ltrs:r < s

(' q < ' s) + (' s < ' r)
apply inl, (strictly_order_preserving _), (transitivity ltqr ltrs). Qed. End rational. Section logic. Context {x : F}.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F

locator x -> locator' x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F

locator x -> locator' x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

locator' x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

forall (q r : Q) (nu : q < r), {| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |} -> ' q < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
forall (q r : Q) (nu : q < r), ~ {| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |} -> x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

forall (q r : Q) (nu : q < r), {| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |} -> ' q < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r

{| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |} -> ' q < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r

is_inl (l q r nu) -> ' q < x
apply un_inl.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

forall (q r : Q) (nu : q < r), ~ {| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |} -> x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r

~ {| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |} -> x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r

~ is_inl (l q r nu) -> x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
ltqx:' q < x

~ is_inl (inl ltqx) -> x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
l0:x < ' r
~ is_inl (inr l0) -> x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
ltqx:' q < x

~ is_inl (inl ltqx) -> x < ' r
simpl; intros f; destruct (f tt).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r
l0:x < ' r

~ is_inl (inr l0) -> x < ' r
intros ?; assumption. Defined.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F

locator' x -> locator x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F

locator' x -> locator x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r

(' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
yes:locates_right l' nu

(' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
no:~ locates_right l' nu
(' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
yes:locates_right l' nu

(' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
yes:locates_right l' nu

' q < x
exact (locates_right_true l' nu yes).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
no:~ locates_right l' nu

(' q < x) + (x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
q, r:Q
nu:q < r
no:~ locates_right l' nu

x < ' r
exact (locates_right_false l' nu no). Defined. End logic. Section logic2. Context {x : F}. Coercion locator_locator' : locator >-> locator'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

locator'_locator l = l
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

locator'_locator l = l
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q

locator'_locator l q = l q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q

locator'_locator l q r = l q r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r

locator'_locator l q r nu = l q r nu
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r

match dec (locates_right {| locates_right := fun (q r : Q) (nu : q < r) => {| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |}; locates_right_true := fun (q r : Q) (nu : q < r) => un_inl (l q r nu); locates_right_false := fun (q r : Q) (nu : q < r) => match l q r nu as s return (~ is_inl s -> x < ' r) with | inl _ => fun f : ~ Unit => match f tt return (x < ' r) with end | inr b => fun _ : ~ is_inl (inr b) => b end |} nu) with | inl a => inl (locates_right_true {| locates_right := fun (q r : Q) (nu : q < r) => {| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |}; locates_right_true := fun (q r : Q) (nu : q < r) => un_inl (l q r nu); locates_right_false := fun (q r : Q) (nu : q < r) => match l q r nu as s return (~ is_inl s -> x < ' r) with | inl _ => fun f : ~ Unit => match f tt return (x < ' r) with end | inr b => fun _ : ~ is_inl (inr b) => b end |} nu a) | inr b => inr (locates_right_false {| locates_right := fun (q r : Q) (nu : q < r) => {| dhprop_hprop := Build_HProp (is_inl (l q r nu)); dec_dhprop := decidable_is_inl (l q r nu) |}; locates_right_true := fun (q r : Q) (nu : q < r) => un_inl (l q r nu); locates_right_false := fun (q r : Q) (nu : q < r) => match l q r nu as s return (~ is_inl s -> x < ' r) with | inl _ => fun f : ~ Unit => match f tt return (x < ' r) with end | inr b0 => fun _ : ~ is_inl (inr b0) => b0 end |} nu b) end = l q r nu
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
nu:q < r

match dec (is_inl (l q r nu)) with | inl a => inl (un_inl (l q r nu) a) | inr b => inr (match l q r nu as s return (~ is_inl s -> x < ' r) with | inl _ => fun f : ~ Unit => match f tt return (x < ' r) with end | inr b0 => fun _ : ~ Empty => b0 end b) end = l q r nu
destruct (l q r nu); auto. Qed. Definition locsig : _ <~> locator' x := ltac:(issig).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x

locator'_locator l' = l'
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x

locator'_locator l' = l'
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
p:locsig^-1 (locator'_locator l') = locsig^-1 l'

locator'_locator l' = l'
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
locsig^-1 (locator'_locator l') = locsig^-1 l'
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x
p:locsig^-1 (locator'_locator l') = locsig^-1 l'

locator'_locator l' = l'
refine (equiv_inj (locsig ^-1) p).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x

locsig^-1 (locator'_locator l') = locsig^-1 l'
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l':locator' x

(fun (q r : Q) (nu : q < r) => {| dhprop_hprop := Build_HProp (is_inl (locator'_locator l' q r nu)); dec_dhprop := decidable_is_inl (locator'_locator l' q r nu) |}; fun (q r : Q) (nu : q < r) => un_inl (locator'_locator l' q r nu); fun (q r : Q) (nu : q < r) => match locator'_locator l' q r nu as s return (~ is_inl s -> x < ' r) with | inl _ => fun f : ~ Unit => match f tt return (x < ' r) with end | inr b => fun _ : ~ Empty => b end) = (locates_right l'; locates_right_true l'; locates_right_false l')
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r), locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r), ~ locates_right0 q r nu -> x < ' r

(fun (q r : Q) (nu : q < r) => {| dhprop_hprop := Build_HProp (is_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end); dec_dhprop := decidable_is_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end |}; fun (q r : Q) (nu : q < r) => un_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end; fun (q r : Q) (nu : q < r) => match match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end as s return (~ is_inl s -> x < ' r) with | inl _ => fun f : ~ Unit => match f tt return (x < ' r) with end | inr b => fun _ : ~ Empty => b end) = (locates_right0; locates_right_true0; locates_right_false0)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r), locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r), ~ locates_right0 q r nu -> x < ' r

(fun (q r : Q) (nu : q < r) => {| dhprop_hprop := Build_HProp (is_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end); dec_dhprop := decidable_is_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end |}) = locates_right0
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r), locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r), ~ locates_right0 q r nu -> x < ' r
q, r:Q
nu:q < r

{| dhprop_hprop := Build_HProp (is_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end); dec_dhprop := decidable_is_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end |} = locates_right0 q r nu
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r), locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r), ~ locates_right0 q r nu -> x < ' r
q, r:Q
nu:q < r

is_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end = locates_right0 q r nu
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
locates_right0:forall q r : Q, q < r -> DHProp
locates_right_true0:forall (q r : Q) (nu : q < r), locates_right0 q r nu -> ' q < x
locates_right_false0:forall (q r : Q) (nu : q < r), ~ locates_right0 q r nu -> x < ' r
q, r:Q
nu:q < r

is_inl match dec (locates_right0 q r nu) with | inl a => inl (locates_right_true0 q r nu a) | inr b => inr (locates_right_false0 q r nu b) end = is_inl (dec (locates_right0 q r nu))
destruct (dec (locates_right0 q r nu)); auto. Qed. Definition equiv_locator_locator' : locator x <~> locator' x := equiv_adjointify locator_locator' locator'_locator locator'_locator_locator' locator_locator'_locator.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r

~ (' q < x) -> locates_left l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r

~ (' q < x) -> locates_left l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
f:locates_right l' ltqr -> ' q < x

~ (' q < x) -> locates_left l' ltqr
exact (not_contrapositive f). Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r

x < ' q -> locates_left l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r

x < ' q -> locates_left l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
ltxq:x < ' q

locates_left l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
ltxq:x < ' q

~ (' q < x)
apply lt_flip; assumption. Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r

~ (x < ' r) -> locates_right l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r

~ (x < ' r) -> locates_right l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
nltxr:~ (x < ' r)

locates_right l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
nltxr:~ (x < ' r)

~~ locates_right l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
nltxr:~ (x < ' r)
f:~ locates_right l' ltqr -> x < ' r

~~ locates_right l' ltqr
exact (not_contrapositive f nltxr). Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r

' r < x -> locates_right l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r

' r < x -> locates_right l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
ltrx:' r < x

locates_right l' ltqr
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
q, r:Q
l':locator' x
ltqr:q < r
ltrx:' r < x

~ (x < ' r)
apply lt_flip; assumption. Qed. End logic2.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
q:Q
eps:Qpos Q

q - ' eps < q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
q:Q
eps:Qpos Q

q - ' eps < q
apply (pos_minus_lt_compat_r q ('eps)), eps. Qed. Local Open Scope mc_scope.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
q:Q
eps:Qpos Q

q < q + ' eps
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
q:Q
eps:Qpos Q

q < q + ' eps
apply (pos_plus_lt_compat_r q ('eps)), eps. Qed. Section bounds. (* Given a real with a locator, we can find (integer) bounds. *) Context {x : F} (l : locator x). Definition ltN1 (q : Q) : q - 1 < q := ltQnegQ q 1. Definition P_lower (q : Q) : Type := locates_right l (ltN1 q).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
k:Q

IsHProp (P_lower k)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
k:Q

IsHProp (P_lower k)
apply _. Qed. Definition ltxN1 : x - 1 < x := (fst (pos_minus_lt_compat_r x 1) lt_0_1).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

hexists (fun q : Q => P_lower q)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

hexists (fun q : Q => P_lower q)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

hexists (fun q : Q => ' q < x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
hexists (fun q : Q => P_lower q)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

hexists (fun q : Q => ' q < x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x - 1 < ' q < x)

hexists (fun q : Q => ' q < x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x - 1 < ' q < x)
hex':{q : Q & x - 1 < ' q < x}

hexists (fun q : Q => ' q < x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x - 1 < ' q < x)
hex':{q : Q & x - 1 < ' q < x}

{q : Q & ' q < x}
destruct hex' as [q [ltx1q ltqx]]; exists q; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)

hexists (fun q : Q => P_lower q)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
hqlt':{q : Q & ' q < x}

hexists (fun q : Q => P_lower q)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
q:Q
lt:' q < x

hexists (fun q : Q => P_lower q)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
q:Q
lt:' q < x

{q : Q & P_lower q}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
q:Q
lt:' q < x

P_lower q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun q : Q => ' q < x)
q:Q
lt:' q < x

locates_right l (ltN1 q)
apply ltrx_locates_right; assumption. Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

{q : Q & ' q < x}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

{q : Q & ' q < x}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
qP_lower:{q : Q & P_lower q}

{q : Q & ' q < x}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
Pq:P_lower q

{q : Q & ' q < x}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
Pq:P_lower q

' (q - 1) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
Pq:locates_right l (ltN1 q)

' (q - 1) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q:Q
Pq:is_inl (l (q - 1) q (ltN1 q))

' (q - 1) < x
apply (un_inl _ Pq). Qed. Definition lt1N (r : Q) : r < r + 1 := ltQposQ r 1. Definition P_upper (r : Q) : DHProp := locates_left l (lt1N r).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
k:Q

IsHProp (P_upper k)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
k:Q

IsHProp (P_upper k)
apply _. Qed. Definition ltx1N : x < x + 1 := (fst (pos_plus_lt_compat_r x 1) lt_0_1).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

hexists (fun r : Q => P_upper r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

hexists (fun r : Q => P_upper r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

hexists (fun r : Q => x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
hexists (fun r : Q => P_upper r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

hexists (fun r : Q => x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x < ' q < x + 1)

hexists (fun r : Q => x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x < ' q < x + 1)
hex':{q : Q & x < ' q < x + 1}

hexists (fun r : Q => x < ' r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hex:hexists (fun q : Q => x < ' q < x + 1)
hex':{q : Q & x < ' q < x + 1}

{r : Q & x < ' r}
destruct hex' as [r [ltxr ltrx1]]; exists r; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)

hexists (fun r : Q => P_upper r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
hqlt':{r : Q & x < ' r}

hexists (fun r : Q => P_upper r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
r:Q
lt:x < ' r

hexists (fun r : Q => P_upper r)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
r:Q
lt:x < ' r

{r : Q & P_upper r}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
r:Q
lt:x < ' r

P_upper r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy canonical_names.lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
hqlt:hexists (fun r : Q => x < ' r)
r:Q
lt:x < ' r

locates_left l (lt1N r)
apply ltxq_locates_left; assumption. Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

{r : Q & x < ' r}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

{r : Q & x < ' r}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
rP_upper:{r : Q & P_upper r}

{r : Q & x < ' r}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
Pr:P_upper r

{r : Q & x < ' r}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
Pr:P_upper r

x < ' (r + 1)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
Pr:locates_left l (lt1N r)

x < ' (r + 1)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
Pr:~ is_inl (l r (r + 1) (lt1N r))

x < ' (r + 1)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:' r < x
Pr:~ is_inl (inl l0)

x < ' (r + 1)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:x < ' (r + 1)
Pr:~ is_inl (inr l0)
x < ' (r + 1)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:' r < x
Pr:~ is_inl (inl l0)

x < ' (r + 1)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:' r < x
Pr:~ Unit

x < ' (r + 1)
destruct (Pr tt).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
r:Q
l0:x < ' (r + 1)
Pr:~ is_inl (inr l0)

x < ' (r + 1)
assumption. Qed. Instance inc_N_Q : Cast nat Q := naturals_to_semiring nat Q. Instance inc_fin_N {n} : Cast (Fin n) nat := fin_to_nat.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n

0 < ' epsilon / 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n

0 < ' epsilon / 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n

0 < ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
0 < / 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n

0 < ' epsilon
apply epsilon.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n

0 < / 3
apply pos_dec_recip_compat, lt_0_3.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3

0 < 3 / ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
lt0':0 < 3 / ' epsilon
{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3

0 < 3 / ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3

0 < 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
0 < / ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3

0 < 3
apply lt_0_3.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3

0 < / ' epsilon
apply pos_dec_recip_compat, epsilon.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
lt0':0 < 3 / ' epsilon

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0:0 < ' epsilon / 3
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

r < q + ' n * ' epsilon / 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

r < q + ' n * ' epsilon / 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

r < ' n * ' epsilon / 3 + q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

r - q < ' n * ' epsilon / 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

(r - q) * (3 / ' epsilon) < ' n * ' epsilon / 3 * (3 / ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

3 / ' epsilon * (r - q) < ' n * ' epsilon / 3 * (3 / ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

3 / ' epsilon * (r - q) < ' n * (' epsilon / 3) * (3 / ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

3 / ' epsilon * (r - q) < ' n * (' epsilon / 3 * (3 / ' epsilon))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

3 / ' epsilon * (r - q) < ' n * (' epsilon * (/ 3 * (3 / ' epsilon)))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

3 / ' epsilon * (r - q) < ' n * (' epsilon * (/ 3 * 3 / ' epsilon))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

3 / ' epsilon * (r - q) < ' n * (' epsilon * (3 / 3 / ' epsilon))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

3 / ' epsilon * (r - q) < ' n * (' epsilon * (1 / ' epsilon))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0

3 / ' epsilon * (r - q) < ' n * (' epsilon / ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
apepsilon0:' epsilon <> 0

3 / ' epsilon * (r - q) < ' n * (' epsilon / ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
apepsilon0:' epsilon <> 0

3 / ' epsilon * (r - q) < ' n * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
apepsilon0:' epsilon <> 0

3 / ' epsilon * (r - q) < ' n
assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q

forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q

forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

grid (fin_incl k) < grid (fsucc k)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' ((fin_to_nat k).+1)%nat - 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' ((' k).+1)%nat = ' (' k) + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' (' k) + 1 - 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' ((' k).+1)%nat = ' (' k) + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' (' k + 1) = ' (' k) + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' (' k) + ' 1 = ' (' k) + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' (' k) + 1 = ' (' k) + 1
reflexivity.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' (' k) + 1 - 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' (' k) + 1 - 1 = ' (' k) - 1 + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' (' k) - 1 + 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' (' k) + 1 - 1 = ' (' k) - 1 + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' (' k) + (1 - 1) = ' (' k) - 1 + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' (' k) + (-1 + 1) = ' (' k) - 1 + 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

' (' k) - 1 + 1 = ' (' k) - 1 + 1
reflexivity.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2

q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' (' k) - 1 + 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1

q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' (' k) - 1 + 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1

(' (' k) - 1) * (' epsilon / 3) < (' (' k) - 1 + 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
lt2:(' (' k) - 1) * (' epsilon / 3) < (' (' k) - 1 + 1) * (' epsilon / 3)
q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' (' k) - 1 + 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1

(' (' k) - 1) * (' epsilon / 3) < (' (' k) - 1 + 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1

' (' k) - 1 < ' (' k) - 1 + 1
apply lt1.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
lt2:(' (' k) - 1) * (' epsilon / 3) < (' (' k) - 1 + 1) * (' epsilon / 3)

q + (' fin_to_nat k - 1) * (' epsilon / 3) < q + (' (' k) - 1 + 1) * (' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
k:Fin n.+2
lt1:' (' k) - 1 < ' (' k) - 1 + 1
lt2:(' (' k) - 1) * (' epsilon / 3) < (' (' k) - 1 + 1) * (' epsilon / 3)

(' fin_to_nat k - 1) * (' epsilon / 3) < (' (' k) - 1 + 1) * (' epsilon / 3)
exact lt2.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

P fin_zero
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

P fin_zero
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

' grid (fsucc fin_zero) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

' (q + (' (' fsucc fin_zero) - 1) * (' epsilon / 3)) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

' (q + (' fin_to_nat (fsucc fin_zero) - 1) * (' epsilon / 3)) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

' (q + (' 1%nat - 1) * (' epsilon / 3)) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

' (q + (1 - 1) * (' epsilon / 3)) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

' (q + 0 * (' epsilon / 3)) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

' (q + 0) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp

' q < x
assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

~ P fin_last
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

~ P fin_last
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' grid (fin_incl fin_last)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + (' (' fin_incl fin_last) - 1) * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + (' fin_to_nat (fin_incl fin_last) - 1) * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + (' (n.+1)%nat - 1) * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + (' (n + 1) - 1) * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + (' n + ' 1 - 1) * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + (' n + 1 - 1) * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + (' n + (1 - 1)) * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + (' n + 0) * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + ' n * (' epsilon / 3))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' (q + ' n * ' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
' r < ' (q + ' n * ' epsilon / 3)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

x < ' r
exact ltxr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero

' r < ' (q + ' n * ' epsilon / 3)
apply strictly_order_preserving; try trivial.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:P (fin_incl u)
Pltxueps:~ P (fsucc u)

{u : Q & ' u < x < ' (u + ' epsilon)}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:P (fin_incl u)
Pltxueps:~ P (fsucc u)

' grid (fin_incl (fin_incl u)) < x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))

' grid (fin_incl (fin_incl u)) < x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))

' grid (fin_incl (fin_incl u)) < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))
x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))

' grid (fin_incl (fin_incl u)) < x
apply (locates_right_true l (lt_grid (fin_incl u)) Pltux).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
H:Funext
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
ltqx:' q < x
r:Q
ltxr:x < ' r
n:nat
lt3rqn:3 / ' epsilon * (r - q) < naturals_to_semiring nat Q n
lt0':0 < 3 / ' epsilon
ap30:(3 : Q) <> 0
ltn3eps:r < q + ' n * ' epsilon / 3
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
P:=fun k : Fin n.+2 => locates_right l (lt_grid k):Fin n.+2 -> DHProp
left_true:P fin_zero
right_false:~ P fin_last
u:Fin n.+1
Pltux:locates_right l (lt_grid (fin_incl u))
Pltxueps:~ locates_right l (lt_grid (fsucc u))

x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:(3 : Q) <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))

x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:(3 : Q) <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3) : Q:Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, grid (fin_incl k) < grid (fsucc k)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u)) Pltxueps:x < ' grid (fsucc (fsucc u))

x < ' (grid (fin_incl (fin_incl u)) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u)) Pltxueps:x < ' (q + (' (' fsucc (fsucc u)) - 1) * (' epsilon / 3))

x < ' (q + (' (' fin_incl (fin_incl u)) - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u)) Pltxueps:x < ' (q + (' (' fsucc (fsucc u)) - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat (fin_incl (fin_incl u)) - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u)) Pltxueps:x < ' (q + (' (' fsucc (fsucc u)) - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:=locates_right_false l (lt_grid (fsucc u)) Pltxueps:x < ' (q + (' fin_to_nat (fsucc (fsucc u)) - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' ((fin_to_nat u).+2)%nat - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' (fin_to_nat u + 1 + 1) - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' (fin_to_nat u + 1) + ' 1 - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u + ' 1 + ' 1 - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u + 1 + 1 - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u + 2 - 1) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u + (2 - 1)) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u + (-1 + 2)) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1 + 2) * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + ((' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3)))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3)) < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3) < q + (' fin_to_nat u - 1) * (' epsilon / 3) + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 * (' epsilon / 3) < ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 * ' epsilon / 3 < ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

' epsilon * 2 / 3 < ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

' epsilon * 1 * 2 / 3 < ' epsilon * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

' epsilon * (1 * 2) / 3 < ' epsilon * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

' epsilon * 2 / 3 < ' epsilon * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

' epsilon * (2 / 3) < ' epsilon * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

PropHolds (0 < ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))
2 / 3 < 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

PropHolds (0 < ' epsilon)
apply epsilon.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 / 3 < 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 / 3 * 3 < 1 * 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 * (/ 3 * 3) < 1 * 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 * (3 / 3) < 1 * 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 * 1 < 1 * 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 < 1 * 3
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
cast_pres_ordering:StrictlyOrderPreserving qinc
x:F
l:locator x
epsilon:Qpos Q
q:Q
n:nat
ap30:3 <> 0
grid:=fun k : Fin n.+3 => q + (' (' k) - 1) * (' epsilon / 3):Fin n.+3 -> Q
lt_grid:forall k : Fin n.+2, q + (' (' fin_incl k) - 1) * (' epsilon / 3) < q + (' (' fsucc k) - 1) * (' epsilon / 3)
u:Fin n.+1
Pltxueps:~ locates_right l (lt_grid (fsucc u))
ltxbla:x < ' (q + (' fin_to_nat u - 1) * (' epsilon / 3) + 2 * (' epsilon / 3))

2 < 3
exact lt_2_3. Qed. End bounds. Section arch_struct. Context {x y : F} (l : locator x) (m : locator y) (ltxy : x < y). Definition P (qeps' : Q * Qpos Q) : Type := match qeps' with | (q' , eps') => (prod (locates_left l (ltQnegQ q' eps')) (locates_right m (ltQposQ q' eps'))) end.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
qeps':(Q * Qpos Q)%type

IsHProp (P qeps')
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
qeps':(Q * Qpos Q)%type

IsHProp (P qeps')
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps':Qpos Q

IsHProp (P (q, eps'))
apply istrunc_prod. Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
qeps':(Q * Qpos Q)%type

Decidable (P qeps')
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
qeps':(Q * Qpos Q)%type

Decidable (P qeps')
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps':Qpos Q

Decidable (P (q, eps'))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps':Qpos Q

Decidable (locates_left l (ltQnegQ q eps') * locates_right m (ltQposQ q eps'))
apply _. Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q

s < t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q

s < t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q

s < t
refine (strictly_order_reflecting _ _ _ ltst').
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

hexists P
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

{x : _ & P x}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

P (q, epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

locates_left l (ltQnegQ q epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
locates_right m (ltQposQ q epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

locates_left l (ltQnegQ q epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

x < ' (q - ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

q - ' epsilon = s
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
x < ' s
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

q - ' epsilon = s
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

(t + s) / 2 - (t - s) / 2 = s
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

s + (t - s) / 2 - (t - s) / 2 = s
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

s + ((t - s) / 2 - (t - s) / 2) = s
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

s + 0 = s
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

s = s
reflexivity.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

x < ' s
assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

locates_right m (ltQposQ q epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

' (q + ' epsilon) < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

q + ' epsilon = t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q
' t < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

q + ' epsilon = t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

(t + s) / 2 + (t - s) / 2 = t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

t - (t - s) / 2 + (t - s) / 2 = t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

t + (- ((t - s) / 2) + (t - s) / 2) = t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

t + 0 = t
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

t = t
reflexivity.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hs:hexists (fun q : Q => x < ' q < y)
s:Q
ltxs:x < ' s
ltsy:' s < y
ht:hexists (fun q : Q => ' s < ' q < y)
t:Q
ltst':' s < ' t
ltty:' t < y
q:=(t + s) / 2:Q
ltst:s < t
epsilon:=Qpos_diff s t ltst / 2:Qpos Q

' t < y
assumption. Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

{q : Q & x < ' q < y}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

{q : Q & x < ' q < y}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

{x : _ & P x}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
R:{x : _ & P x}
{q : Q & x < ' q < y}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

{x : _ & P x}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

nat <~> Q * Qpos Q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
forall x : Q * Qpos Q, IsHProp (P x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
forall x : Q * Qpos Q, Decidable (P x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
hexists (fun x : Q * Qpos Q => P x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

nat <~> Q * Qpos Q
apply QQpos_eq.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

forall x : Q * Qpos Q, IsHProp (P x)
apply P_isHProp.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

forall x : Q * Qpos Q, Decidable (P x)
apply P_dec.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y

hexists (fun x : Q * Qpos Q => P x)
apply P_inhab.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
R:{x : _ & P x}

{q : Q & x < ' q < y}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
R:{qeps' : Q * Qpos Q & (locates_left l (ltQnegQ (fst qeps') (snd qeps')) * locates_right m (ltQposQ (fst qeps') (snd qeps')))%type}

{q : Q & x < ' q < y}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)

{q : Q & x < ' q < y}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)

x < ' q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)
' q < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)

x < ' q
nrefine (locates_right_false l _ lleft).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
ltxy:x < y
q:Q
eps:Qpos Q
lleft:locates_left l (ltQnegQ q eps)
mright:locates_right m (ltQposQ q eps)

' q < y
nrefine (locates_right_true m _ mright). Qed. End arch_struct. Section unary_ops. Context {x : F} (l : locator x).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

locator (- x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x

locator (- x)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r

((' q < - x) + (- x < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q

((' q < - x) + (- x < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x

((' q < - x) + (- x < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)
((' q < - x) + (- x < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x

((' q < - x) + (- x < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x

- x < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x

- ' r < x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltnrx:' (- r) < x

' (- r) < x
assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)

((' q < - x) + (- x < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)

' q < - x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)

x < - ' q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
q, r:Q
ltqr:q < r
ltnrnq:- r < - q
ltxnq:x < ' (- q)

x < ' (- q)
assumption. Qed. Section recip_pos. Context (xpos : 0 < x). Definition recip_nu := positive_apart_zero x xpos.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x

locator (// (x; recip_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x

locator (// (x; recip_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)

locator (// (x; recip_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qzero:q = 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0

' q < // (x; recip_nu)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0

' q < ?Goal1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0
?Goal1 < // (x; recip_nu)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0

' q < ?Goal1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0

q < ?y
exact qneg.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qneg:q < 0

' 0 < // (x; recip_nu)
rewrite preserves_0; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qzero:q = 0

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qzero:q = 0

' q < // (x; recip_nu)
rewrite qzero, preserves_0; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0

r ≶ 0
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0

r ≶ 0
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0

0 < r
apply (transitivity qpos ltqr).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x

// (x; recip_nu) < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r

// (x; recip_nu) < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r

0 < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r
rpos':0 < ' r
// (x; recip_nu) < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r

0 < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r

qinc 0 < ' r
apply strictly_order_preserving; try apply _; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltrrx:' (/ r) < x
rpos:0 < r
rpos':0 < ' r

// (x; recip_nu) < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
rpos':0 < ' r
ltrrx:// (' r; positive_apart_zero (' r) rpos') < x
rpos:0 < r

// (x; recip_nu) < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
rpos':0 < ' r
ltrrx:// (' r; positive_apart_zero (' r) rpos') < x
rpos:0 < r
ltxrr:let apx0 := positive_apart_zero x (transitivity (pos_recip_compat (' r) rpos') ltrrx) in // (x; apx0) < ' r

// (x; recip_nu) < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
rpos':0 < ' r
ltrrx:// (' r; positive_apart_zero (' r) rpos') < x
rpos:0 < r
ltxrr:// (x; positive_apart_zero x (strictorder_trans 0 (// (' r; positive_apart_zero (' r) rpos')) x (pos_recip_compat (' r) rpos') ltrrx)) < ' r

// (x; recip_nu) < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
rpos':0 < ' r
ltrrx:// (' r; positive_apart_zero (' r) rpos') < x
rpos:0 < r
ltxrr:// (x; recip_nu) < ' r

// (x; recip_nu) < ' r
exact ltxrr.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)

((' q < // (x; recip_nu)) + (// (x; recip_nu) < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)

' q < // (x; recip_nu)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)

0 < ' q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
qpos':0 < ' q
' q < // (x; recip_nu)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)

0 < ' q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)

qinc 0 < ' q
apply strictly_order_preserving; try apply _; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
ltxrq:x < ' (/ q)
qpos':0 < ' q

' q < // (x; recip_nu)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
qpos':0 < ' q
ltxrq:x < // (' q; positive_apart_zero (' q) qpos')

' q < // (x; recip_nu)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
qpos':0 < ' q
ltxrq:x < // (' q; positive_apart_zero (' q) qpos')
ltrqx:' q < // (x; positive_apart_zero x xpos)

' q < // (x; recip_nu)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xpos:0 < x
recippos:0 < // (x; recip_nu)
q, r:Q
ltqr:q < r
qpos:0 < q
qap0:q ≶ 0
rap0:r ≶ 0
ltrrrq:/ r < / q
qpos':0 < ' q
ltxrq:x < // (' q; positive_apart_zero (' q) qpos')
ltrqx:' q < // (x; recip_nu)

' q < // (x; recip_nu)
exact ltrqx. Qed. End recip_pos. End unary_ops. Section recip_neg. Context {x : F} (l : locator x) (xneg : x < 0). Definition recip_neg_nu := negative_apart_zero x xneg.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0

locator (// (x; recip_neg_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0

locator (// (x; recip_neg_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x

locator (// (x; recip_neg_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (- (// (- x; recip_nu negxpos)))

locator (// (x; recip_neg_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// negate_apart (- x; recip_nu negxpos))

locator (// (x; recip_neg_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// (- - x; apart_negate (- x) (recip_nu negxpos)))

locator (// (x; recip_neg_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// (x; recip_neg_nu))

locator (// (x; recip_neg_nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// (- - x; apart_negate (- x) (recip_nu negxpos)))
- - x = x
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// (x; recip_neg_nu))

locator (// (x; recip_neg_nu))
assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
xneg:x < 0
negxpos:0 < - x
l':locator (// (- - x; apart_negate (- x) (recip_nu negxpos)))

- - x = x
apply negate_involutive. Qed. End recip_neg. Section unary_ops2. Context {x : F} (l : locator x) (nu : x ≶ 0).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0

locator (// (x; nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0

locator (// (x; nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xneg:x < 0

locator (// (x; nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xpos:0 < x
locator (// (x; nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xneg:x < 0

locator (// (x; nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xneg:x < 0
l':=locator_recip_neg l xneg:locator (// (x; recip_neg_nu xneg))

locator (// (x; nu))
rewrite (recip_proper_alt x x (negative_apart_zero x xneg) nu) in l'; try reflexivity; exact l'.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xpos:0 < x

locator (// (x; nu))
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x:F
l:locator x
nu:x ≶ 0
xpos:0 < x
l':=locator_recip_pos l xpos:locator (// (x; recip_nu xpos))

locator (// (x; nu))
rewrite (recip_proper_alt x x (positive_apart_zero x xpos) nu) in l'; try reflexivity; exact l'. Qed. End unary_ops2. Section binary_ops. Context {x y : F} (l : locator x) (m : locator y). (** TODO the following two should be proven in Classes/orders/archimedean.v *) Context (char_plus_left : forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => (' s < x) /\ (' (q - s) < y))) (char_plus_right : forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => (x < ' t) /\ (y < ' (r - t)))).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)

locator (x + y)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)

locator (x + y)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

' q - ' s < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

' q - ' s < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

' q - ' (q - u) < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

' q - (' q + ' (- u)) < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

' q + (- ' q - ' (- u)) < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

' q - ' q - ' (- u) < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

0 - ' (- u) < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

- ' (- u) < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

- - ' u < y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q

' u < y
assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x

' q < x + y
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x

hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltsx:' s < x

' (q - s) < y
rewrite preserves_minus; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)

((' q < (x + y)%mc) + ((x + y)%mc < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)

x + y < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)

hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)

{t : Q & ((x < ' t) * (y < ' (r - t)))%type}
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

((x < ' t) * (y < ' (r - t)))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

y < ' (r - t)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

r - (q - u + (r - q) / 2) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q
y < ' (u + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

r - (q - u + (r - q) / 2) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

r - (q - u + ' epsilon) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

r + (- (q - u) - ' epsilon) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

r + (- q + u - ' epsilon) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

r + (u - q - ' epsilon) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

r + (u - q) - ' epsilon = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

r + u - q - ' epsilon = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + r - q - ' epsilon = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (r - q) - ' epsilon = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (r - q - ' epsilon) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (- q + r - ' epsilon) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (- q + (r - ' epsilon)) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (- q + (r + q) / 2) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (- q + (q + (r - q) / 2)) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (- q + q + (r - q) / 2) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (0 + (r - q) / 2) = u + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

u + (r - q) / 2 = u + ' epsilon
reflexivity.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 2:Qpos Q
X:q + ' epsilon = r - ' epsilon
u:Q
ltuy:' u < y
ltyuepsilon:y < ' (u + ' epsilon)
s:=q - u:Q
qsltx:' q - ' s < y
sltseps:s < s + ' epsilon
ltxseps:x < ' (s + ' epsilon)
t:=s + ' epsilon:Q

y < ' (u + ' epsilon)
assumption. Qed. (* TODO construct locators for multiplications. *)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)

locator (x * y)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)

locator (x * y)
Abort.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)

locator (x ⊓ y)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)

locator (x ⊓ y)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r

((' q < x ⊓ y) + (x ⊓ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltqy:' q < y

((' q < x ⊓ y) + (x ⊓ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltyr:y < ' r
((' q < x ⊓ y) + (x ⊓ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltqy:' q < y
((' q < x ⊓ y) + (x ⊓ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltyr:y < ' r
((' q < x ⊓ y) + (x ⊓ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltqy:' q < y

((' q < x ⊓ y) + (x ⊓ y < ' r))%type
apply inl, meet_lt_l; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltyr:y < ' r

((' q < x ⊓ y) + (x ⊓ y < ' r))%type
apply inr, meet_lt_r_r; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltqy:' q < y

((' q < x ⊓ y) + (x ⊓ y < ' r))%type
apply inr, meet_lt_r_l; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltyr:y < ' r

((' q < x ⊓ y) + (x ⊓ y < ' r))%type
apply inr, meet_lt_r_r; assumption. Qed.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)

locator (x ⊔ y)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)

locator (x ⊔ y)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r

((' q < x ⊔ y) + (x ⊔ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltqy:' q < y

((' q < x ⊔ y) + (x ⊔ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltyr:y < ' r
((' q < x ⊔ y) + (x ⊔ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltqy:' q < y
((' q < x ⊔ y) + (x ⊔ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltyr:y < ' r
((' q < x ⊔ y) + (x ⊔ y < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltqy:' q < y

((' q < x ⊔ y) + (x ⊔ y < ' r))%type
apply inl, join_lt_l_l; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltqx:' q < x
ltyr:y < ' r

((' q < x ⊔ y) + (x ⊔ y < ' r))%type
apply inl, join_lt_l_l; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltqy:' q < y

((' q < x ⊔ y) + (x ⊔ y < ' r))%type
apply inl, join_lt_l_r; assumption.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
x, y:F
l:locator x
m:locator y
char_plus_left:forall (q : Q) (x y : F), ' q < x + y <-> hexists (fun s : Q => ((' s < x) * (' (q - s) < y))%type)
char_plus_right:forall (r : Q) (x y : F), x + y < ' r <-> hexists (fun t : Q => ((x < ' t) * (y < ' (r - t)))%type)
q, r:Q
ltqr:q < r
ltxr:x < ' r
ltyr:y < ' r

((' q < x ⊔ y) + (x ⊔ y < ' r))%type
apply inr, join_lt_r; assumption. Qed. End binary_ops. Section limit. Context {xs : nat -> F}. Context {M} {M_ismod : CauchyModulus Q F xs M}. Context (ls : forall n, locator (xs n)).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F

IsLimit Q F xs l -> locator l
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F

IsLimit Q F xs l -> locator l
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l

locator l
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r

((' q < l) + (l < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

((' q < l) + (l < ' r))%type
(* TODO we are doing trisection so we have the inequality: *)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

q + ' epsilon < r - ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
((' q < l) + (l < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

q + ' epsilon < r - ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

q + ' epsilon + ' epsilon < r - ' epsilon + ' epsilon
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

q + ' epsilon + ' epsilon < r + (- ' epsilon + ' epsilon)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

q + ' epsilon + ' epsilon < r + 0
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

q + ' epsilon + ' epsilon < r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

q + (' epsilon + ' epsilon) < r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

- q + (q + (' epsilon + ' epsilon)) < - q + r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

- q + q + (' epsilon + ' epsilon) < - q + r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

' epsilon + ' epsilon < - q + r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

' epsilon + ' epsilon < r - q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

' epsilon * 1 + ' epsilon * 1 < r - q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

' epsilon * 2 < r - q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

(r - q) / 3 * 2 < r - q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

(r - q) * (/ 3 * 2) < r - q
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

(fun q0 : Q => (r - q) * (/ 3 * 2) < q0) (r - q)
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q

(r - q) * (/ 3 * 2) < (r - q) * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q

(r - q) * (/ 3 * 2) < (r - q) * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q

/ 3 * 2 < 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q

3 * (/ 3 * 2) < 3 * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q

3 / 3 * 2 < 3 * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q

1 * 2 < 3 * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q
3 <> 0
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q

1 * 2 < 3 * 1
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q

2 < 3
exact lt_2_3.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
rqpos:0 < r - q

3 <> 0
apply apart_ne, positive_apart_zero, lt_0_3.
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon

((' q < l) + (l < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' (q + ' epsilon) < xs (M (epsilon / 2))

((' q < l) + (l < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' (r - ' epsilon)
((' q < l) + (l < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' (q + ' epsilon) < xs (M (epsilon / 2))

((' q < l) + (l < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' (q + ' epsilon) < xs (M (epsilon / 2))

' q < l
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' q + ' (' epsilon) < xs (M (epsilon / 2))

' q < l
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' q + ' (' epsilon) < xs (M (epsilon / 2))
ltqxseps:' q < xs (M (epsilon / 2)) - ' (' epsilon)

' q < l
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltqepsxs:' q + ' (' epsilon) < xs (M (epsilon / 2))
ltqxseps:' q < xs (M (epsilon / 2)) - ' (' epsilon)

xs (M (epsilon / 2)) - ' (' epsilon) < l
apply (modulus_close_limit _ _ _ _ _).
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' (r - ' epsilon)

((' q < l) + (l < ' r))%type
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' (r - ' epsilon)

l < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' r + ' (- ' epsilon)

l < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' r - ' (' epsilon)

l < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' r - ' (' epsilon)
ltxsepsr:xs (M (epsilon / 2)) + ' (' epsilon) < ' r

l < ' r
Q:Type
Aap:Apart Q
Aplus:Plus Q
Amult:Mult Q
Azero:Zero Q
Aone:One Q
Aneg:Negate Q
Arecip:DecRecip Q
Ale:Le Q
Alt:Lt Q
U:RationalsToField Q
Qrats:Rationals Q
Qdec_paths:DecidablePaths Q
Qtriv:TrivialApart Q
Trichotomy0:Trichotomy lt
F:Type
Alt0:Lt F
Ale0:Le F
Aap0:Apart F
Azero0:Zero F
Aone0:One F
Aplus0:Plus F
Anegate:Negate F
Amult0:Mult F
Arecip0:Recip F
Ajoin:Join F
Ameet:Meet F
Forderedfield:OrderedField F
Fabs:Abs F
Farchimedean:ArchimedeanProperty Q F
Fcomplete:IsComplete Q F
Qroundup:RoundUpStrict Q
H:Funext
H0:Univalence
Q_eq:nat <~> Q
QQpos_eq:nat <~> Q * Qpos Q
cast_pres_ordering:StrictlyOrderPreserving qinc
qinc_strong_presving:IsSemiRingStrongPreserving qinc
xs:nat -> F
M:Qpos Q -> nat
M_ismod:CauchyModulus Q F xs M
ls:forall n : nat, locator (xs n)
l:F
islim:IsLimit Q F xs l
q, r:Q
ltqr:q < r
epsilon:=Qpos_diff q r ltqr / 3:Qpos Q
ltqepsreps:q + ' epsilon < r - ' epsilon
ltxsreps:xs (M (epsilon / 2)) < ' r - ' (' epsilon)
ltxsepsr:xs (M (epsilon / 2)) + ' (' epsilon) < ' r

l < xs (M (epsilon / 2)) + ' (' epsilon)
apply (modulus_close_limit _ _ _ _ _). Qed. End limit. End locator.